Here's the story, with the usual caveats that it's all fantasy and completely unrealistic, a fairy tale with a logical/philosophical point.

A group of 100 superb logicians and philosophers—to save pronoun complexity, we assume all are male—are being held captive on an island by an evil tyrant. In response to pressure from civil rights groups, the tyrant announces that he will set up a night court to which any prisoner can go and ask to be released. Any petitioner who has green eyes will be released; all other petitioners will be thrown into a volcano. Now, in fact all 100 prisoners have green eyes. But they are not allowed access to any reflective device that will tell them the color of their own eyes, and although they can see one another, they are forbidden to communicate among themselves in any way. As a concession, the civil rights groups may make one statement to all the prisoners, but it must be a statement whose truth is already known to all of them.

What statement should they make in order to get the prisoners released through the night court?

Spoiler:

The answer is this: "There is at least one prisoner who has green eyes." While this statement certainly fulfills the restriction, since each of the prisoners can see 99 others with green eyes, it takes some thinking to see how that gets them all off. It turns out that they have to use all their philosophical/logical powers to make use of it, as will be explained below. Two interesting facts are highlighted by this puzzle: (1) it is possible to communicate new knowledge to people without stating any facts they didn't already know; (2) the fact that other minds exist and think in the same way as one's own mind may have important consequences.

To see the explanation, we first need to consider what I shall call the 2-person game. That is, we assume there are only two prisoners on the island, both having green eyes. Obviously each of them knows what I shall call the Fundamental Fact, namely that there is at least one green-eyed prisoner. What each of them doesn't know in advance is that THE OTHER PRISONER ALSO KNOWS THAT FACT. Prisoner A doesn't know he has green eyes, and therefore cannot assume that prisoner B knows the Fundamental Fact, and vice versa. Once the statement is made and certified, however, A knows that B knows and B knows that A knows the fundamental fact. Being logicians of a high order, each of them knows that, if he personally does NOT have green eyes, then the other now knows himself to be green-eyed and will surely petition for release that night. And if he does have green eyes, the other will not be sure and will not petition for release. When morning arrives and neither has been released, both will know that they have green eyes and will petition for release on the second night.

Now add a third person, getting the 3-person game. For clarity, the prisoners are A, B, and C. A reasons that if he does NOT have green eyes, then that fact is known to B and C, and they will then interpret the Fundamental Fact as applying to just the two of them. That is, they will be playing the two-person game. In that case, they will petition for release on the second night. On the other hand, if A DOES have green eyes, B and C will not interpret the Fundamental Fact that way and will not petition for release on the second night. The same reasoning will be engaged in by B and C. If, on the third morning, each of them finds he is not alone, he will know that he has green eyes, and so all three will petition for release on the third night.

The induction should now be obvious. All 100 brilliant logician/philosophers will watch for the morning of the 100th day. Each of them, waking up that day and finding that he is not alone, will conclude that he has green eyes, and all 100 will petition for release that night.

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."

I suppose we have to assume the Civil Rights Group can see all the eyes?

"but it must be a statement whose truth is already known to all of them. " /// ....already known to each of them.

Real Name: bobbo the existential pragmatic evangelical anti-theist and Class Warrior.Asking: What is the most good for the most people?Sample Issue: Should the Feds provide all babies with free diapers?

Actually, Bobbo gave you a better answer than I did. The statement must not convey any fact not already known to the prisoners. None of them KNOWS that each of them can see 99 green eyes. So what you suggested violates the conditions of the problem.

Yes, if you want a different, easier puzzle, you can just have the people say, "There are at least 99 prisoners with green eyes." Then each prisoner will say, "If I didn't have green eyes, the others would all know instantly that they DO have green eyes, because they'd only see 98 prisoners with green eyes and therefore conclude that they must be the 99th." That wouldn't violate the conditions of the puzzle, but it would not be the puzzle stated. It would be easier.

If you want an easier puzzle, fine, but why object to one that isn't as easy as what you thought of? One can either play the game or argue about the rules. You take your choice: Which do you prefer?

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."

The terminology of "easier" puzzle is a puzzle to me. Thats the whole point of a puzzle: to solve it as simply as possible. 2+2=4 NOT 2+2=5+3 divided by 2. Do you mean to say something else?

Real Name: bobbo the existential pragmatic evangelical anti-theist and Class Warrior.Asking: What is the most good for the most people?Sample Issue: Should the Feds provide all babies with free diapers?

bobbo_the_Pragmatist wrote:The terminology of "easier" puzzle is a puzzle to me. Thats the whole point of a puzzle: to solve it as simply as possible. 2+2=4 NOT 2+2=5+3 divided by 2. Do you mean to say something else?

No, I mean "easier." With the alternative statement, the solution of the puzzle takes only two lines to write out. Compare that with my exposition of the solution of the original puzzle, which I think is about as short as one can be and still be clear at each step. The number of steps required to solve the original puzzle is much larger than in the suggested alternative. And, as you noted, the actual suggested alternative violates the rules in the original puzzle, so it's a different puzzle entirely, and trivially easy, even easier than the alternative as I rephrased it, which is consistent with the rules, and still easier than the original.

The point is, that Austin Harper didn't solve the puzzle. He changed it to a different puzzle. Mathematicians try not to do that. If the mountain is too steep to climb, you keep trying as long as you can. You can always give up and stroll up a nearby hill, but then you can't claim you actually climbed the mountain.

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."

Real Name: bobbo the existential pragmatic evangelical anti-theist and Class Warrior.Asking: What is the most good for the most people?Sample Issue: Should the Feds provide all babies with free diapers?

OlegTheBatty wrote:There is no such thing as an evil tyrant who locks up 100 philosophers. Tyrant, yes . . .

....a Statement against self interest? Very rare.

Real Name: bobbo the existential pragmatic evangelical anti-theist and Class Warrior.Asking: What is the most good for the most people?Sample Issue: Should the Feds provide all babies with free diapers?

OlegTheBatty wrote:There is no such thing as an evil tyrant who locks up 100 philosophers. Tyrant, yes . . .

Technically, you can have a person who is a tyrant simply because they seized power without due process. They wouldn't have to be evil so to do. Another definition of a tyrant is just an absolute ruler who is not bound by any laws. They, too, would not need to be evil.

Using simple statistics, any Logician seeing a 0.99 correlation between being highly logical and green eyed would consider it highly likely that he is green-eyed, too.No extra information required.

Last edited by ElectricMonk on Mon Jan 22, 2018 12:45 pm, edited 1 time in total.

I've come up with a set of rules that describe our reactions to technologies:

Spoiler:

1. Anything that is in the world when you’re born is normal and ordinary and is just a natural part of the way the world works.2. Anything that's invented between when you’re fifteen and thirty-five is new and exciting and revolutionary and you can probably get a career in it.3. Anything invented after you're thirty-five is against the natural order of things.- Douglas Adams

tyrant: A cruel and oppressive dictator. The due process deal is from ancient Greece....currently, NA.

Real Name: bobbo the existential pragmatic evangelical anti-theist and Class Warrior.Asking: What is the most good for the most people?Sample Issue: Should the Feds provide all babies with free diapers?

A tyrant (Greek τύραννος, tyrannos), in the modern English usage of the word, is an absolute ruler unrestrained by law or person, or one who has usurped legitimate sovereignty. Often described as a cruel character, a tyrant defends his position by oppressive means, tending to control almost everything in the state....

This discussion is getting more interesting than I expected it to be. First, a mea culpa: I should have made the statement from the civil-rights group part of the problem. The actual puzzle should have been to explain why that statement enabled the prisoners to get out. Austin Harper immediately picked up on that fact and suggested another way to answer the question that was posed as the puzzle. Now let's make that statement itself the puzzle that is to be explained. We'll try three variants.

1. "Each of you can see 99 green-eyed people." (Austin Harper's suggestion.) In that case, each of the prisoners realizes immediately that he must have green eyes. Otherwise, the other prisoners would be seeing only 98 green-eyed people. All of them will then get released on the first night. (As I mentioned, this does state a fact not known to the prisoners before. None of them knew that the others were also seeing 99 green-eyed people.)

2. "At most one of you is NOT green-eyed." (My variant of what Austin Harper suggested.) In that case each of them will reason: If I didn't have green-eyes, the others would know that I am the exception, and they would get released on the first night. But none could be sure until the following morning, so all 100 would get released on the second night.

3. "At least one of you IS green-eyed." That's the variant I used, and the explanation is as I gave it. The prisoners get released on the 100th night.

Further reflection: When there are only two prisoners, statements 2 and 3 are equivalent. And they do convey information to each of the prisoners. Each of them knew this fact before, but now each knows that the other also knows it.

Last edited by Upton_O_Goode on Mon Jan 22, 2018 12:44 pm, edited 1 time in total.

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."

A tyrant (Greek τύραννος, tyrannos), in the modern English usage of the word, is an absolute ruler unrestrained by law or person, or one who has usurped legitimate sovereignty. Often described as a cruel character, a tyrant defends his position by oppressive means, tending to control almost everything in the state....

Again, it's "often" described as a cruel character, and not "always".

You are correct. In his History of Western Philosophy, Bertrand Russell noted that tyranny originally meant simply having a ruler whose claim to power was not hereditary. But of course, the word acquired its modern meaning of arbitrary, capricious and cruel government long ago. ("Taxation without representation is tyranny"---ascribed to Benjamin Franklin, I believe.)

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."

I admit I don't get it. From the perspective of one prisoner, how does the statement, "There is at least one prisoner who has green eyes," in combination with his observance that his 99 cellmates have green eyes, lead him to the conclusion that he, too, has green eyes?

Isn't that the same as saying, "I've rolled 99 dice and they've all come up ones. Therefore, the probability that the 100th dice will also be a one is certain." But it's not certain. The probability remains one in six, since it's not affected by any other dice that have been thrown.

So, how does the observable fact that his 99 cellmates have green eyes (along with the statement) preclude him having brown, blue, black, grey, hazel, golden, or violet eyes?

"Science is a way of thinking much more than it is a body of knowledge."—Carl Sagan

"Every philosophy is tinged with the coloring of some secret imaginative background, which never emerges explicitly into its train of reasoning."—Alfred North Whitehead

"Knowledge belongs to humanity, and is the torch which illuminates the world."—Louis Pasteur

Nikki Nyx wrote:I admit I don't get it. From the perspective of one prisoner, how does the statement, "There is at least one prisoner who has green eyes," in combination with his observance that his 99 cellmates have green eyes, lead him to the conclusion that he, too, has green eyes?

Isn't that the same as saying, "I've rolled 99 dice and they've all come up ones. Therefore, the probability that the 100th dice will also be a one is certain." But it's not certain. The probability remains one in six, since it's not affected by any other dice that have been thrown.

So, how does the observable fact that his 99 cellmates have green eyes (along with the statement) preclude him having brown, blue, black, grey, hazel, golden, or violet eyes?

It works by induction on the number of prisoners. In the case of two prisoners, even though each one knows there is at least one prisoner with green eyes (because he can see that the other guy has green eyes), neither of them knows that the OTHER one is seeing a green-eyed prisoner; hence neither can be sure the other guy knows there is at least one green-eyed prisoner. It is not the fact stated but the fact that it WAS stated to both of them that gives each of them the additional information that "the other guy knows there is a green-eyed prisoner." Hence, each reasons, "If I DON'T have green eyes, the other guy knows he is the one who DOES and he'll get released tonight. When both are still present the next morning, each knows that the other guy wasn't sure of being the green-eyed one, and hence they must both be green-eyed and both will apply for release on the second night.

Then, for three prisoners, each, starting with the assumption that he himself does NOT have green eyes, deduces that the other two would be in exactly the situation just described for two prisoners, and both would therefore get released on the second night. When all three are present on the third morning, they all realize that they all have green eyes, and apply for release on the third night. And so on....

This is pure logic. Probability doesn't enter into it. That fact is emphasized by the condition that the prisoners have to bet their lives on being green-eyed when they apply for release.

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."

Nikki Nyx wrote:I admit I don't get it. From the perspective of one prisoner, how does the statement, "There is at least one prisoner who has green eyes," in combination with his observance that his 99 cellmates have green eyes, lead him to the conclusion that he, too, has green eyes?

Isn't that the same as saying, "I've rolled 99 dice and they've all come up ones. Therefore, the probability that the 100th dice will also be a one is certain." But it's not certain. The probability remains one in six, since it's not affected by any other dice that have been thrown.

So, how does the observable fact that his 99 cellmates have green eyes (along with the statement) preclude him having brown, blue, black, grey, hazel, golden, or violet eyes?

It works by induction on the number of prisoners. In the case of two prisoners, even though each one knows there is at least one prisoner with green eyes (because he can see that the other guy has green eyes), neither of them knows that the OTHER one is seeing a green-eyed prisoner; hence neither can be sure the other guy knows there is at least one green-eyed prisoner. It is not the fact stated but the fact that it WAS stated to both of them that gives each of them the additional information that "the other guy knows there is a green-eyed prisoner." Hence, each reasons, "If I DON'T have green eyes, the other guy knows he is the one who DOES and he'll get released tonight. When both are still present the next morning, each knows that the other guy wasn't sure of being the green-eyed one, and hence they must both be green-eyed and both will apply for release on the second night.

Then, for three prisoners, each, starting with the assumption that he himself does NOT have green eyes, deduces that the other two would be in exactly the situation just described for two prisoners, and both would therefore get released on the second night. When all three are present on the third morning, they all realize that they all have green eyes, and apply for release on the third night. And so on....

This is pure logic. Probability doesn't enter into it. That fact is emphasized by the condition that the prisoners have to bet their lives on being green-eyed when they apply for release.

I discovered why I didn't get it. The original problem didn't specify that the night court was an ongoing thing, so I presumed it was a one-time thing, that the prisoners had only one opportunity to petition for release. Obviously, the induction wouldn't work in that case, since it hinges on observing the behavior of the other prisoners over time. Phew! I'm relieved that I'm not suffering from early onset dementia.

"Science is a way of thinking much more than it is a body of knowledge."—Carl Sagan

"Every philosophy is tinged with the coloring of some secret imaginative background, which never emerges explicitly into its train of reasoning."—Alfred North Whitehead

"Knowledge belongs to humanity, and is the torch which illuminates the world."—Louis Pasteur

Nikki Nyx wrote: I discovered why I didn't get it. The original problem didn't specify that the night court was an ongoing thing, so I presumed it was a one-time thing, that the prisoners had only one opportunity to petition for release. Obviously, the induction wouldn't work in that case, since it hinges on observing the behavior of the other prisoners over time. Phew! I'm relieved that I'm not suffering from early onset dementia.

You're not. I'll certify that.

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."

Nikki raises a good point. The puzzle would remain the same if the tyrant announced: "Any prisoner or group of prisoners may request a hearing at any time. When such a request is made, a night session of court will be held on the day of the request, and those eligible for release will be released. THIS IS A ONE-TIME OFFER. ONLY ONE SUCH COURT SESSION WILL BE HELD."

"Relieve yourself of the illusion that you're writing for the ages. The ages will decide who is doing that on their own; you don't get a vote."